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Erschienen in:
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2019 | OriginalPaper | Buchkapitel

1. Elementary Particle Dynamics

verfasst von : Oliver M. O’Reilly

Erschienen in: Engineering Dynamics

Verlag: Springer International Publishing

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Abstract

In this first chapter, we cover the basics on kinematics and kinetics of particles with particular emphasis on the Cartesian coordinate system. Euler’s first law (also known as Newton’s second law or the balance of linear momentum) is used to relate the kinematics of the particle to the forces acting on it. This law provides a set of differential equations for the motion of the particle. We illuminate the developments using three ubiquitous examples involving projectile motion both in the presence and absence of drag forces. Our treatment of dynamics makes extensive use of vector calculus. For the interested student, a summary of the needed results from vector calculus is presented in Appendix A.

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Nächstes Kapitel Particles and Cylindrical Polar Coordinates
Fußnoten
1
It is necessary to use a dummy variable \(\tau \) as opposed to the variable t when evaluating this integral because we are integrating the magnitude of the velocity as \(\tau \) varies between \(t_0\) and t. If we take the derivative with respect to t of the integral, then, using the fundamental theorem of calculus, we would find, as expected, that \(\dot{s}(t) = \left| \!\left| {\mathbf {v}}\right| \!\right| \). Had we not used the dummy variable \(\tau \) but rather t to perform the integration, then the derivative of the resulting integral with respect to t would not yield \(\dot{s}(t) = \left| \!\left| {\mathbf {v}}(t)\right| \!\right| \).
 
2
Here, we are invoking the inverse function theorem of calculus. If \(\dot{s}\) were zero, then the particle would be stationary (i.e., s would be constant), but time would continue increasing, so there would not be a one-to-one correspondence between s and t.
 
3
In other words, the more problems one examines, the better.
 
4
Later on, we hope sooner rather than later, you should revisit this problem using the cylindrical polar coordinates, \(r = R_0\) and \(\theta = \omega t\), and establish the forthcoming results using cylindrical polar basis vectors. This coordinate system is discussed in Chapter 2.
 
5
Leonhard Euler (1707–1783) made enormous contributions to mechanics and mathematics. We follow C. Truesdell (see Essays II and V in [97]) in crediting \(\mathbf{F}\) = \(m\mathbf{a}\) to Euler. As noted by Truesdell, these differential equations can be seen in a 1749 paper by Euler [31, Pages 101–105]. Truesdell’s essays also contain copies of certain parts of a related seminal paper [32] by Euler that was published in 1752.
 
6
Isaac Newton (1642–1727) wrote his second law in Volume 1 of his famous Principia in 1687 as follows: The change of motion is proportional to the motive force impressed; and is made in the direction of the right line in which that force is impressed. (Cf. [67, Page 13].)
 
7
George Gabriel Stokes (1819–1903) made several seminal contributions to mechanics and mathematics. These contributions include Stokes’ theorem in calculus and the Navier–Stokes equations in fluid dynamics. For a perspective on Stokes’ work and the research on drag that followed, we highly recommend Vecsey and Goldenfeld’s paper [99].
 
8
This speed is equal to 99.09 miles per hour.
 
Metadaten
Titel
Elementary Particle Dynamics
verfasst von
Oliver M. O’Reilly
Copyright-Jahr
2019
Buch
Engineering Dynamics

Print ISBN: 978-3-030-11744-3

Electronic ISBN: 978-3-030-11745-0

Copyright-Jahr: 2019

DOI
https://doi.org/10.1007/978-3-030-11745-0_1

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